reflected lines KI and CI, and directed to the given point I, is given; hence AEO is given in position. Join OC, and the angle ECO, being equal (I. 8. El.) to CEO, is given; and consequently CO, and the centre O, are given.

COMPOSITION.

Make (VI. 3. El.) AE: BH:: AD: BD, join DC, make BH: AE :: DC: DK; and, from the points K and C, inflect KI and CI, by Note XXIV. such that GE shall tend to D, produce AE and CO, making the angle ECO equal to CEO; the intersection O is centre of the required circle.

For join AG, CF, OB, and BH. Because AE, or AG: BH, or BF :: AD: BD, and the triangles ADG and BDF have a common angle at D, they are (VI. 15. El.) similar; consequently AD: BD :: DG : DF :: DK: DC, and IG is parallel to FC; and therefore the circles touch at E. But the triangle, BFH, having its sides BF and BH parallel to AG and AE, the sides of the isosceles triangle GAE, must likewise be isosceles ; wherefore the circles meet at F: And, since BH is parallel to EO,

Again, the angle ECO being

they must touch at that point. equal to CEO, the side OE is equal to OC; and consequently the circle described from O, and which touches at E and F, must also pass through C.

Note XXVI.-Page 404.

The French philosophers have, at the instance of Borda, lately proposed and adopted the centesimal division of the quadrant, as easier, more consistent, and better adapted to our scale of arithmetic. On that basis, they have also constructed their ingenious system of measures. The distance of the Pole from the Equator was determined with the most scrupulous accuracy, by a chain of triangles extending from Calais to Barcelona, and since prolonged to the

Balearic Isles. Of this quadrantal arc, the ten millionth part, or the tenth part of a second, constitutes the metre, or unit of linear extension. From the metre again, are derived the several measures of surface and of capacity; and water, at its greatest contraction, furnishes the standard of weights.

It would be most desirable, if this elegant and universal system were adopted, at least in books of science. Whether, with all its advantages, it be ever destined to obtain a general currency in the ordinary affairs of life, seems extremely questionable. At all events, its reception must necessarily be very slow and gradual; and, in the meantime, this innovation is productive of much inconvenience, since it not only deranges our habits, but tends to displace our delicate instruments and elaborate tables. The fate of the centesimal division may finally depend on the continued merit of the works framed after that model,

Note XXVII.-Page 406.

The 24th Proposition of the 3d Book of the Elements affords a very simple expression for the sum of the sines of progressive arcs. Suppose the diameter AO were drawn; then BE+CF +DG = HG = HO+DO, or 2S,AB+2S,AC+2S,AD= HO+S,AD, and C,AB + S,AC+S,AD=4H0+1S,AD= AO.T,BAO+S,AD. Wherefore, in general, S,a+S,2a+ S,3a.....S,na VS,na. Cot,a+S,na. Hence the sum of the sines in the whole semicircle is Cot,a. Thus, if the sines for each degree up to 180°, the radius being unit, were added together, the amount would be 114,58866.

Note XXVIII.-Pages 407, 408.

I tis hence easy to determine the relative variation of an arc te its sine and cosine. Let a express the small increment which the

a

arc A receives; then, the radius being assumed equal to unit, S‚Ã+α = S,A.Cos,α + Cos,A.S,, and Cos, A + a Cos, A. Cos,α-S,A. S,α. But the sine of a may be considered as equal to that arc itself, and its cosine differs not sensibly from the radius. Wherefore, S,A+a S,A+a. Cos,A, and Cos, A Cos,A-a.S,A; consequently, 1st, the increment of the sine of A, or S,A+ — S,A,a. Cos, A; and, 2d, the decrement of the cosine of A, or Cos,A — Cos,A+α, = .S,A.

=

THE general properties of the sines of compound arcs may be derived with great facility from Prop. 24. of Book VI. of the Elements. For, since AB.CD + BC.AD=AC.BD, it is evident that AB.CD+BC.AD=AC.¿BD; but (cor. 1. def. Trig.) the semichord of an arc is the same as the sine of half the arc, and consequently, by substitution, S, AB x S,CD + S, BC × S, ABCD = S, ABC x S,BCD. Let AB=L, BC=M, and CD=N; wherefore ABCD =L+M+N, ABC=L+M, and BCD=M+N, and hence the general result: S,LXS,N+S,M×S, L+M+N=SL+M× S,M+N, This expression, in which L, M, and N are any arcs whatever. variously transformed, will exhibit all the theorems respecting

sines.

1. Put A=M, B-N, and let L be the complement of A. Then, Cos,AX S,B+S,A× S,A+B+90°-A=S,90°-A+A XS,A+B; that is, since the sine of an arc increased by a quadrant is the same as its cosine, S,A x Cos,B+ Cos,A x S,B =RXS,A+B.

2. Let the arc B be taken on the opposite side, or substitute

-B for it in the last expression, and S,A x Cos,B-Cos,A x S,B =RXS,A-B.

2. In art. 1., for A substitute its complement; then S,A+B =S,90°—A+B=S, 90° +A-B= Cos,A-B, and hence

Cos,AX Cos,B+S,A×S,B=RX Cos,A-B.

4. In art. 2. likewise, substitute for A its complement, and the result will become Cos, A x Cos, B-S,A x S,B= Rx Cos,A+B.

5. In art. 1., let A=B, and 2S,A x Cos, A=RXS,2A.

6. In art. 4. let A=B, and Cos',A-S',A=RX Cos,2A.

7. Suppose LA—B, and M=N=B; then the general expression becomes S,A-B x S,B,+ S‚B × S,A—B + 2B = S,ABB x S,2B, or S,B (S‚A + B + S‚A—B) = S,A XS,2B.

8. Since, from art. 5., Rx S,2B S,B x 2 Cos, B, therefore, by combining this with the last result, R(S,A+B+S,A—B)= S,A X 2Cos, B.

9. In the preceding article, for A substitute its complement, and R(S,90°—A+B+S,90°— A— B) = Cos,A × 2Cos,B, or R(Cos,A+B= Cos,A-B) = Cos,A x 2 Cos, B.

10. In art. 8. change A and B for their complements, and R(S,180°-A-B+S, -A+B) = Cos, A x 2S,B, or R(S,A+B—S,A—B) = Cos,A × 2S,B.

11. In art. 9. likewise, change A and B for their complements;

4

then R(Cos,180°— A − B + Cos, —A+B=S,A+2S,B, or R(Cos,A—B—Cos,A+B) = S,A×2S,B.

12. In art. 10. transform A and B into A + B and A-B, and consequently, for A+B and A-B, substitute 2A and 2B; then R (S,2A — S,2B) = Cos, A + B + S‚A—B, or R(S,2A-S,2B)=Cos,A+B×S,A-B.

13. Make the same transformation in article 11., and R(Cos,2B-Cos,2A) = S‚A+B× 2S,A—B, or R(Cos, 2B-Cos,2A)=S,A+BXS,A-B.

14. Lastly, suppose LNB, and M-A-B; then the general expression becomes, S',B+S,A — B × S‚A+B = S3‚A, or S,A+B x S‚A—B = S1‚A-S',B. But, by the preceding article, R (Cos,2B — Cos,2A) = S‚A + B × S‚A—B; whence R(Cos,2B-Cos,2A)=S',A-S',B.

Note XXXI.-Page 414.

The general expression for the sine of the multiple arc was obtained by mere induction; but this mode of inference, in most cases so convenient, is perhaps not quite satisfactory. A complete investigation may be derived from the Theory of Func

tions.

On inspecting the successive formation of the sines of the multiple arcs, it appears, 1. That the odd powers only of s occur; 2. That the coefficient of the first term is always n, and the other coefficients are its functions of third, fifth, &c. orders; and 3. That since, in the case when n=1, the rest of the coefficients evidently vanish, those coefficients in general, as affected by opposite signs, must in each term produce a mutual balance.